Chapter 11.1, Problem 46E

### Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550

Chapter
Section

### Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550
Textbook Problem

# Determine whether sequence converges or diverges. If it converges, find the limit.46. an = 2−ncos nπ

To determine

Whether the sequence converges or diverges and obtain the limit if the sequence converges.

Explanation

Given:

The sequence is an=2ncosnπ (or) an=cosnπ2n .

Definition used:

If an is a sequence and limnan exists, then the sequence an is said to be converges; otherwise it diverges.

Theorem used: Squeeze Theorem:

If xnznyn for nN and limnxn=limnyn=L then the value of limnzn is L.

Calculation:

Obtain the limit of the sequence to investigate whether the sequence converges or diverges.

Compute the value of limnan=limncosnπ2n .

Since 1cosnπ1 and divide by 2n , apply the Squeeze Theorem and obtain the relation as follows:

12ncosnπ2n12n

limn(12n)limn(cosnπ2n)limn(12n) (1)

Obtain the value of limn(12n)

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